Complete Study Notes on Mechanical Vibrations and Degrees of Freedom

Fundamentals and Theory of Vibration

• Definition of Vibration: Vibration, also referred to as oscillation, is defined as any motion that repeats itself after a specific interval of time. Typical examples provided include the swinging of a pendulum and the motion of a plucked string.

• Theory of Vibration: This field involves the study of the oscillatory motions of bodies and the specific forces associated with these motions.

• The Physical Process of Vibration:

  • When a body is displaced by an external force, internal forces emerge in the form of elastic energy.

  • These internal forces attempt to return the body to its original state (equilibrium).

  • At the equilibrium position, the accumulated elastic energy is converted entirely into kinetic energy.

  • This kinetic energy causes the body to move in the opposite direction, and the process repeats cyclically.

The Importance and Impact of Vibration Study

• Human Physiological Involvement:

  • Hearing: The eardrums must vibrate to process sound.

  • Sight: Light waves undergo vibration.

  • Breathing: Associated with the vibration of the lungs.

  • Locomotion: Walking involves the periodic oscillatory motion of the hands and legs.

  • Speech: Requires the oscillatory motion of the larynges and tongues.

• Engineering Failures and Challenges:

  • Unbalance in Engines: Most prime movers face vibration issues due to inherent unbalance caused by faulty design or poor manufacturing.

  • Ground Nuisance: Imbalance in diesel engines can create ground waves powerful enough to be a nuisance in urban environments.

  • Locomotive Safety: The wheels of some locomotives can lift more than $1\,cm$ off the track at high speeds due to imbalance.

  • Turbine Failures: Vibrations in turbines cause spectacular mechanical failures, specifically in the blades and disks.

  • Material Fatigue: Cyclic variation of induced stress from vibrations leads to material fatigue and eventual failure of structures supporting centrifugal or reciprocating machines (steam/gas engines, pumps, motors).

  • Wear and Noise: Vibration causes rapid wear in parts like bearings and gears and generates excessive noise.

  • Fasteners: In machines, vibration can loosen nuts and other fasteners.

  • Manufacturing Quality: In metal cutting, vibration causes "chatter," resulting in poor surface finish.

• Resonance:

  • Definition: This phenomenon occurs when the natural frequency of a machine or structure coincides with the frequency of external excitation.

  • Impact: It leads to excessive deflections and catastrophic system failures.

• Human-System Integration: Transmission of vibration to humans causes discomfort and logic/efficiency loss. Vibration of instrument panels can make meters difficult to read or cause malfunctions.

Beneficial Applications of Vibration

While often detrimental, vibration is utilized profitably in many sectors:

• Industrial Equipment: Vibratory conveyors, hoppers, sieves, compactors, and pile driving.

• Consumer Products: Washing machines, electric toothbrushes, clocks, and electric massaging units.

• Medical/Professional Tools: Dentist's drills.

• Manufacturing Enhancement: Improving efficiency in machining, casting, forging, and welding processes.

• Specialized Processes: Vibratory finishing of materials and electronic circuits used to filter out unwanted frequencies.

• Research: Simulating earthquakes for geological research and conducting studies for nuclear reactor design.

Technical Definitions in Vibration Analysis

1. Periodic Motion: Motion that repeats itself at equal intervals of time.

2. Time Period ($T$): The time taken to complete exactly one cycle.

3. Frequency ($f$): The number of cycles completed per unit of time.

4. Simple Harmonic Motion (SHM): A periodic motion of particles where acceleration is always directed toward the mean position.

5. Amplitude: The maximum displacement of a vibrating body from its mean (equilibrium) position.

6. Free Vibrations: The vibration of a system due to its own elastic properties without any external exciting force.

7. Natural Frequency: The specific frequency at which a system undergoes free vibration.

8. Resonance: A state where the frequency of external force matches the natural frequency of the system.

9. Damping: The resistance to the motion of a vibrating body.

10. Degree of Freedom (DOF): The number of independent coordinates required to specify the complete configuration of a system at any given instant.

Elementary Parts of a Vibratory System

A vibratory system consists of four idealized elements:

• Mass or Inertia ($m$):

  • Assumed to be a rigid body.

  • Gains or loses kinetic energy ($KE$) based on velocity changes.

  • According to Newton's law: force equals the product of mass and acceleration ($F = m \times a$).

  • Work done on the mass is transformed into kinetic energy.

• Spring ($k$):

  • Possesses elasticity and is assumed to have negligible mass.

  • Force exists only if there is relative displacement between the two ends (extension or compression).

  • Work done in deforming the spring is stored as potential energy (strain energy).

  • Linear Spring: Follows Hooke's Law, where spring force is proportional to deformation. The constant of proportionality is called stiffness or the spring constant ($k$).

• Damper ($c$):

  • Has neither mass nor elasticity.

  • Damping force exists only if there is relative motion between the damper ends.

  • Energy input is converted into heat (non-conservative element).

  • Linear Damping (Viscous Damping): Damping force is proportional to velocity. The coefficient is $c$, measured in force per unit velocity.

  • Nonlinear Damping: For example, frictional drag in fluid is approximately proportional to velocity squared ($v^2$).

• Excitation:

  • The means by which energy enters the system.

  • Can be an external force ($F(t)$) applied to the mass or a motion imparted to the support.

Classification of Vibration

• Free vs. Forced Vibration:

  • Free: System vibrates on its own after an initial disturbance (e.g., a simple pendulum).

  • Forced: System is subjected to a repeating external force (e.g., diesel engines).

• Undamped vs. Damped Vibration:

  • Undamped: No energy is lost due to friction or resistance.

  • Damped: Energy is dissipated. Consideration of damping is critical when analyzing systems near resonance.

• Linear vs. Nonlinear Vibration:

  • Linear: All components (mass, spring, damper) behave linearly. The principle of superposition applies.

  • Nonlinear: Any component behaves nonlinearly. Superposition is not valid. All systems tend toward nonlinearity as the amplitude of oscillation increases.

• Deterministic vs. Random Vibration:

  • Deterministic: The value of excitation is known at any given time.

  • Random (Nondeterministic): Excitation value cannot be predicted at a given time but may exhibit statistical regularity (e.g., wind velocity, road roughness, earthquake ground motion). Response is described via statistical quantities (mean, mean square values).

Detailed Analysis of Degrees of Freedom (DOF)

• Definition: The number of independent spatial coordinates needed to define the geometric location of all masses in a system.

• Formula: $DOF = \text{Number of spatial coordinates} - \text{Number of equations of constraint}$

• Rigid Body in Space: Requires six coordinates (3 rectilinear, 3 angular).

• One-Degree-of-Freedom (1-DOF) Examples:

  • Vertical Spring-Mass System: Mass constrained to move vertically requires only coordinate $x(t)$.

  • Torsional Pendulum: Configuration specified by angular coordinate $\theta(t)$.

  • Mass-Spring-Cantilever: If the cantilever mass is neglected, it acts as a spring element combined with mass $m$.

  • Mass-Pulley-Spring: If the cord is inextensible and does not slip, linear displacement $x(t)$ and angular rotation $\theta(t)$ are dependent; only one coordinate is needed.

  • Flyball Governor: Vibratory motion expressed by angular coordinate $\theta(t)$.

  • Simple Pendulum in XY Plane: Coordinates $x(t)$ and $y(t)$ are related by the constraint $x^2 + y^2 = L^2$. Only one coordinate is independent.

• Two-Degree-of-Freedom (2-DOF) Examples:

  • Two-Spring-Two-Mass System: Requires $x_1(t)$ and $x_2(t)$.

  • Multi-axis Spring-Mass: A mass oscillating along the spring axis AND swinging side-to-side.

  • Pendulum in Space: Described by $\theta(t)$ and $\phi(t)$. While $x, y, z$ coordinates exist, they are restricted by $x^2 + y^2 + z^2 = L^2$.

Sample Problems and Solutions

• Problem 1: Rigid bar system.

  • Question: Determine DOF and specify generalized coordinates for a rigid bar.

  • Solution: Because the bar is rigid, it has 1 DOF. The generalized coordinate is $\theta$, representing angular displacement clockwise from equilibrium.

• Problem 2: Mechanical system with mass center displacement.

  • Question: Determine DOF for a bar where $x$ is displacement of the mass center.

  • Solution: Knowing $x$ is not sufficient for all particles on the bar. Adding clockwise angular rotation $\theta$ allows for the displacement of the right end to be calculated as $x + (\frac{L}{2}) \theta$. Thus, it has 2 DOF.

• Problem 3: Multiple disk and block system.

  • Question: Determine DOF for a system with two disks (centers $O_1, O_2$) and blocks $A, B, C$.

  • Solution: The system has 4 DOF. Coordinates: $\theta_1$ (disk 1), $\theta_2$ (disk 2), $x_1$ (block B download), and $x_2$ (block C downward). Note: Block A is not independent because its motion is $r_1 \theta_1$.

Assessment Questions

• Question A: Give two examples each of the bad and the good effects of vibration.

• Question B: What are the three elementary parts of a vibrating system?

• Question C: Define the number of degrees of freedom of a vibrating system.

• Question D: What is the difference between a discrete and a continuous system? Is it possible to solve any vibration problem as a discrete one?

• Question E: In vibration analysis, can damping always be disregarded?